The following handle holds various files of this Leiden University dissertation:
http://hdl.handle.net/1887/81579
Author: Vendel, E.
Title: Prediction of spatial-temporal brain drug distribution with a novel mathematical
model
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Introduction
Pharmacology is the science that focuses on the interaction between a phar-macological compound (drug) and a living system (e.g. human or rat).
Mathematical pharmacology has gained increasing popularity over the last
years. For example, so-called pharmacokinetic/pharmacodynamic (PKPD) modelling is a popular technique that uses mathematics to describe the time course of drug concentration within the body (pharmacokinetics) and relate it to drug effect (pharmacodynamics). The term mathematical pharmacol-ogy may be intriguing to people not familiar with the field: mathematics and pharmacology seem like two entirely different disciplines. In many cases, however, mathematics is of great value to pharmacological research. First, many quantitative data are collected from experimental studies in phar-macology. Mathematics can help to analyse and make sense of the data. Second, mathematical models are a valuable tool for complementing and accelerating pharmacological research.
In essence, a mathematical model is an abstract description of a certain phenomenon. Most phenomena are very complex and therefore mathemati-cal models are always a simplification of reality. Even though a model leaves out certain aspects of the problem under study, it generally increases our understanding: mathematical models help to gain insights in the origins and mechanisms of the phenomenon as they enable separation of complex prob-lems into multiple levels. A few important uses of mathematical modelling are listed below [1]:
• Hypothesis testing.
• Generation of new insights. • Identification of open questions. • Integration of knowledge.
mathemat-lead for future experiments. Moreover, mathematical models may integrate data from several experiments, thereby increasing insight in the studied phenomenon.
Mathematical models may also have several advantages over experiments: • Every variable can be tracked independently as well as in conjunction
with other variables.
• Tissues and organs can be studied that are relatively inaccessible to experiments (e.g. the brain).
• The need for animal experiments may be reduced.
• Mathematical modelling simulations can be easily repeated.
Altogether, mathematical models allow for a thorough and quantitative characterization of the phenomenon under study.
The development of a quantitative and reliable mathematical model is highly challenging. In developing a model, it is important to be able to quantify the following aspects:
1) What is the question? 2) What data are known? 3) Which area is studied? 4) What assumptions are made? 5) What are the initial conditions? 6) What are the boundary conditions?
These aspects will be further explained below.
1. The formulation of a clear question is, like in the design of experiments, very important for the design of a model. With a clear question, the model can be fine-tuned specifically to answer that question. Aspects of the phe-nomenon being of less relevance to the question may be simplified, while aspects that are more relevant to the question can be included in more detail. Imagine a model on drug distribution within the brain fluids. Drug enters the brain fluids either from the blood stream or by direct injection into the brain and then distributes within the brain. When the focus is on the subsequent drug distribution into the brain fluids, drug entrance into the brain can be simplified, while drug distribution within the brain fluids should be described in more detail.
the concentration of drug in the injection fluid, the speed of injection and the solubility of the drug in the fluid.
3. The area to be studied is the area of interest for the phenomenon. In brain cancer, one wishes to study the brain, while in liver cirrhosis, one wishes to study the liver. This is very straightforward. The real question is, however, how exactly is the area to be studied (the domain) defined? Does one wish to study the entire brain, or just the area where the brain tumour is located? What is the shape of the brain tumour?
4. Before defining the initial conditions, boundary conditions, as well as the model itself, assumptions need to be made. Assumptions are an essential part of modelling and are key to the model as reality is too complex to model. Therefore, mathematical models often are a simple, comprehensible representation of reality, for which assumptions are required. Take a model that describes drug transport from the blood into the brain fluids. A pos-sible assumption would be that drug enters the brain fluids in equal extent from all directions. This implies that the permeability of the brain barriers, separating the blood from the brain fluids, is equal everywhere. This is not (completely) as in the real situation, as the brain barriers may be locally more permeable to the drug. When these local variations in permeability are small, however, the assumption that drug enters the brain fluids from all directions in equal extent is a reasonable one. Then, precisely modelling the brain barriers in all their details according to reality is actually a dis-advantage, comparing the effort it takes to the effect it has.
5. Initial conditions describe the state of a system at the start of the study of this system. Many studies focus on the effect of something on a certain phenomenon. This effect can be caused by a wide variety of things, ranging from developmental processes to mutations within cells to the addition of a compound to the system. In all cases, it is important to define the state of the system before something happens. For example, what is the state of the system before cancer cells start to grow and expand? What is the state of the system before a drug is added to it?
dish. Hence, boundary conditions are imposed that describe no movement across the edges of the petri dish. Alternatively, suppose a model describing drug distribution within the brain fluids, where drug exchanges between the blood and the brain fluids by passive transport across the brain barriers. In addition, active transporter proteins move drug from the blood across the brain barriers into the brain fluids. In this case, boundary conditions take into account both active and passive transport across the brain barriers. Models often have either too few parameters, which leads to oversimpli-fication and therefore makes the model not a good representation of reality, or too many parameters, which may hamper proper understanding. The ultimate goal is therefore to generate a model that includes the smallest number of parameters to best describe and predict reality.
1.1 Processes that govern drug distribution within the brain
1.1
Processes that govern drug distribution within the
brain
Drug that is taken by oral administration, which is the most commonly used route of drug administration, is first absorbed into the bloodstream, from where it may also be transported across the BBB into the brain. Within the brain, drug is distributed by transport within the brain fluids, intra-extracellular exchange and drug binding (see Figure 1.1). A sketch of the brain with the blood, brain tissue (containing the brain ECF) and the cerebrospinal fluid (CSF) is shown in Figure 1.2. All transport processes will be briefly described below. A more extensive description is given in Chapter 2.
BBB
BCSFB
Figure 1.2: Structure of the human brain: blood, brain tissue, CSF and the brain barriers. Blood vessels (red) infiltrate the brain tissue (grey) and branch out into
smaller brain capillaries (inset). At the level of the brain capillaries compounds exchange between the blood and the brain tissue through the BBB. The brain tissue (brain parenchyma) contains the brain cells and the brain ECF. The CSF (blue) is located in the sub-arachnoid space (located between the dura mater, a layer of connective tissue surrounding the brain tissue, and the brain tissue), the brain ventricles and the spine. The blood is separated from the CSF by the blood-CSF barrier (BCSFB) and the blood-arachnoid barrier. The brain barriers are indicated by black squares a-c. (a) The BBB is the barrier between the blood in the brain capillaries and the brain tissue. (b) The BCSFB is the barrier between the blood in the capillaries and the CSF in the brain ventricles. (c) The blood-arachnoid barrier is the barrier between the blood in the blood vessels of the dura mater and the CSF in the sub-arachnoid space. (d) Influx of fluid via penetrating arteries and efflux of fluid along a set of veins. Figure 1a-c is adapted from [5] and is licensed under CC BY 4.0. Figure 1d is adapted with permission from [6].
Drug transport from blood plasma into the brain and vice versa
1.1 Processes that govern drug distribution within the brain
[7]. Blood plasma protein binding is relevant to drug distribution as only
unbound drug can pass the membranes to enter other tissues and organs,
like the brain.
Drug transport across the BBB
The BBB is the major barrier of the brain. It acts as a barrier between the blood within the brain capillaries and the brain extracellular space. In order to maintain stability within the brain, it allows only certain molecules, like water, oxygen and certain nutrients, to easily pass through. The brain en-dothelial cells are tightly held together by so-called ‘tight junctions’. A com-pound may cross the BBB by simple passive transport, facilitated transport, vesicular transport and active transport (section 2.2.3.2, Figure 2.6). In sim-ple passive transport, drug may cross the cells (the transcellular route) or via the space between the cells (the paracellular route) by (passive) dif-fusion. A diffusing compound follows a concentration gradient and moves from a site with higher concentration to a site with lower concentration. Hydrophilic compounds, that cannot easily pass the cell membranes, can only enter and leave the brain by paracellular transport, which is limited by the tight junctions. In facilitated transport, diffusion across the BBB is aided by transport proteins. The availability of these helper molecules is limited and saturation of transport proteins may occur at sufficiently high drug concentrations. In vesicular transport, molecules move through vesicles that are formed within the barrier. Finally, in active transport, a drug may move against the concentration gradient, which requires energy. Active transport exists in two directions: from the blood into the brain (ac-tive influx), or ac(ac-tive transport from the brain back into the blood (ac(ac-tive efflux).
Drug distribution within the brain fluids
Intra-extracellular exchange
The brain ECF surrounds the brain cells. The brain cells hinder drug dif-fusion within the brain ECF, which leads to an effective difdif-fusion that is smaller than normal diffusion (in a medium without obstacles). This hin-drance of the cells on brain ECF diffusion is described by the so-called tortuosity. Drugs may also enter the cells by intra-extracellular exchange. This intra-extracellular exchange occurs by transport across the cell mem-branes, which may be by diffusion but also by active transport.
Drug binding
A drug exerts its effect by specific interaction with target sites. At the same time, a drug may interact with non-specific binding sites all over the brain. This involves non-specific interaction with components of the brain. It is only the unbound drug that is available for interaction with target and off-target sites.
Drug metabolism
Metabolic enzymes are located at the brain barriers, in brain ECF, and in brain cells. They may convert inactive drug to its active form or convert active drug to inactive drug. Therefore, they may affect the concentration of (active) drug within the brain and thereby affect drug-target interactions and drug response [8–10].
All of the above-mentioned processes are the result of systems characteris-tics and drug physico-chemical and biological properties. Systems charac-teristics differ between species and subjects and are also affected by food, age and disease. The processes can furthermore differ over space, by differ-ences in properties of the brain (i.e. the concentration of specific binding sites may differ per location) or local disease. Therefore, there is a need for understanding the spatial distribution of a drug within the brain. Mathe-matical models can help in this understanding.
1.2
Models on drug distribution into and within the brain
af-1.2 Models on drug distribution into and within the brain
fecting drug distribution: models on the (spatial) drug distribution within the brain ECF are often coupled to BBB transport, intra-extracellular ex-change, drug binding and drug metabolism. However, none of the models on drug transport into and within the brain ECF offers complete descrip-tions of drug transport into and within the brain, including BBB active transport, drug binding kinetics and the distinction between specific and non-specific binding sites.
Compartmental models provide a simplified representation of drug distri-bution between several ‘compartments’ of the brain. In these models, com-partments represent parts of the brain (like the brain ECF or the brain cells) or the state of a drug (like bound or unbound). Compartmental mod-els generally include multiple processes relevant to drug distribution within the brain, such as BBB transport, intra-extracellular exchange and bind-ing. However, most compartmental models lack the description of the spatial distribution of a drug within the brain (ECF).
A model that integrates BBB transport with the spatial distribution of a drug within the brain ECF and drug binding kinetics leads to a more accurate prediction of drug concentration-time profiles within the brain physiological compartments. An overview of models focusing on drug de-livery by the brain capillary network, on drug transport across the BBB, transport within the brain ECF, intra-extracellular exchange, metabolism and drug binding kinetics, will be given in Chapter 2.
1.2.1 Prediction of local drug concentrations of drug within the brain
using a new mathematical model
In this thesis, we formulate a model that describes the change in concen-tration of a drug within the brain not only over time but also over space. The model consists of a part of the brain tissue (Figure 1.3b). This ‘brain unit’ is the smallest ‘building block’ of the brain in terms of drug distribu-tion. One unit consists of the brain capillaries, containing the blood plasma, surrounding the brain ECF. There, the blood plasma enters the brain cap-illaries through an artery and leaves through a venule. As drug distribution within the brain is complex and many processes are involved, we build the model step by step.
Uout Uin b) c) a) U out Uin Venule Arteriole
Figure 1.3: Development of the brain unit (network) model a) The proof-of-concept 2D brain unit model. A 2D brain ECF domain (blue) is surrounded by the blood plasma (red). b) The 3D brain unit model. A 3D brain ECF domain (blue) and a blood plasma domain (red) are described. c) The 3D brain unit network, consisting of a network of 3D single brain units.
add the brain capillary blood flow and active transport across the BBB, which improves the realistic properties of the model. Finally we introduce a network of 3D brain units (Figure 1.3c and Chapter 5), in which multiple 3D brain units are connected and differences between units can be studied. Below, we summarize the equations that are used to describe drug concen-tration within the brain unit. We start with the basic model description that is used for the proof-of-concept 2D model (section 1.2.1.1.). Then, we summarize the extensions with a blood plasma domain (section 1.2.1.2) and descriptions of active transport (section 1.2.1.3), as have been implemented in the 3D single brain unit model and in the 3D brain unit network. We end with a short description on how the models are implemented numerically. 1.2.1.1 The basic model
The main equation of the basic model describes drug transport within the brain ECF by diffusion and bulk flow and allows for modification to include loss of drug to binding sites or to the blood. We describe the distribution of free drug within the brain ECF using differential equations.
Differential equations describe how a certain quantity changes depending
on parameters and variables, generally over time and/or space. Here, a vari-able changes value, depending on position or time, while parameter values remain constant. Variables and quantities can be dependent or independent. For example, the concentration of a drug (dependent quantity) changes over time (independent variable), while the change in time does not depend on anything.
1.2 Models on drug distribution into and within the brain
change in a quantity (i.e. the concentration of free drug in the brain ECF) in one independent variable (i.e. time or space) on one or multiple param-eters (i.e. the binding rate constant). The solution of the ODE (i.e. the concentration of free drug at a certain time-point) depends on one variable (i.e. time) only. On the contrary, a PDE relates the rate of change of one quantity (i.e. the concentration of free drug in the brain ECF) to at least two independent variables (i.e. time ánd space).
The change in concentration of free drug within the brain ECF over time and space can be described with the following PDE:
Change of free drug over time = Diffusion + Bulk flow − Binding (1.1) The concentration of the diffusing free drug depends on time, but also on the location in the x-,y-, and z-directions. Here, the first right-hand side term describes drug diffusion within the brain ECF. In diffusion, the concen-tration of a compound distributes over space in a time-dependent manner (it takes a certain time until molecules have travelled a certain distance).
Diffusion is quantified by the diffusion coefficient, D. The brain ECF is full
of obstacles, including the cells. Diffusion within the brain ECF is hindered by these obstacles, which leads to an effective diffusion smaller than nor-mal (in a medium without obstacles). This is modelled with the so-called tortuosity λ, thereby dividing the normal diffusion D by λ2, resulting in a
smaller diffusion coefficient [11].
The second right-hand side term describes the transport of drug by brain
ECF bulk flow. We assume that this flow is unidirectional (in the direction
of the CSF).
Finally, the last term on the right-hand side describes binding of drug to
specific and non-specific binding sites. The equation is based on earlier
work by Charles Nicholson, see i.e. [11–13]. This work does not explicitly describe drug transport across the BBB nor the kinetics of drug binding. In our model, we capture the entry and elimination of drug into and from the brain ECF across the BBB with boundary conditions and include drug binding, as is described below.
Drug binding kinetics
obtain the following system of equations:
Change of free drug over time = Diffusion + Bulk flow
Changeof f reedrugovertime =−f(S binding) − g(NS binding) Change of S bound drug over time = f(S binding)
Change of NS bound drug over time = g(NS binding),
(1.2)
where S bound drug is the concentration of drug bound to specific binding sites and NS bound drug is the concentration of drug bound to non-specific binding sites. The functions f(S binding) and g(NS binding) describe the concentrations of drug bound to specific binding sites and drug bound to non-specific binding sites, respectively. Both functions are based on the rate of drug association to binding sites and the rate of drug dissociation from drug-binding site complexes. The total concentrations of binding sites is assumed to be constant. The values of free and bound drug are inter-dependent, which means that the free drug concentration depends on the concentration of bound drug and that the bound drug concentration de-pends on the concentration of free drug.
Boundary conditions
At the boundaries of the brain unit, drug within the blood plasma crosses the BBB to enter and leave the brain ECF. A flux describes the amount of drug transported across the BBB per area per time. The flux results from the concentration difference between the blood plasma and the brain ECF and is described by
Diffusion into brain ECF = P (Free drug(plasma) − Free drug(ECF)), (1.3) where P is the permeability of the BBB to a certain drug, Free drug(plasma) the concentration of free drug within the blood plasma and Free drug(ECF) the concentration of free drug within the brain ECF. If the value of the right hand side is positive, then drug transport is directed towards the brain ECF and if the value is negative, drug transport is directed towards the blood plasma.
Drug concentration within the blood plasma
1.2 Models on drug distribution into and within the brain
concentration is described by the dose divided by the distribution volume. The distribution volume is the theoretical volume that is needed to keep the total amount of drug in the body at the same concentration as in the blood plasma.
The change in concentration of orally administered drug over time is mod-elled as a function that depends on both the rate of absorption of drug into blood plasma and the rate of elimination of drug from blood plasma [7]:
Change of free drug (plasma) over time =Chan
Bioavailability·Dose
Volume of distribution·f(absorption, elimination)
(1.4) Here, f(absorption, elimination) is an exponential decaying function that depends on the rate constant of absorption into the blood plasma and the rate constant of elimination from the blood plasma. For oral administration, an initial increase, followed by a decrease is usual.
1.2.1.2 Extension with a blood plasma domain
We can add an explicit blood plasma domain to the basic model to describe drug exchange from the blood with the brain. To do this, we include the brain capillary blood flow velocity and exchange of drug in blood plasma and brain ECF. In addition to Equation (1.4), we denote:
Change of Free drug(plasma) over time = f(Blood flow),
with f(Blood flow) being the change in concentration of free drug in the blood plasma over time and space as a consequence of the brain capillary blood flow, quantified by the brain capillary blood flow velocity. The brain capillary blood flow is directed from the nearest neighbouring arteriole to the nearest neighbouring venule.
Adding a new blood plasma domain requires the definition of additional boundary conditions for the blood plasma bordering the brain ECF, which are the opposite of boundary conditions (1.3):
Diffusion into plasma..=..P(Free drug(ECF)−Free drug(plasma)). (1.5)
1.2.1.3 Extension with active BBB transport
affinity of a drug for the transporters, which is quantified by the concen-trations of drug at which half of the maximum transport rate is reached, respectively.
1.2.1.4 Model parameters and values
A wide range of parameters (ranging from the brain capillary blood flow velocity to parameters on binding kinetics) is included in the model. The values of the parameters originate from literature, based on which physio-logical ranges for each value are determined. If necessary, data from litera-ture are converted into the right units, since, in all the used equations, the units left of the ‘=’-sign should match the units right of the ‘=’-sign. In certain cases, no data on parameter values are available. In these cases an ‘educated guess’ is made using related parameters that are available. For example, data on non-specific binding kinetics are scarce. However, data on the fraction unbound (the ratio of unbound drug concentration to to-tal drug concentration) are readily available. In that case, the parameters of non-specific binding kinetics are chosen such that the value for fraction unbound predicted by the model matches its value found in literature. 1.2.1.5 Simulating the model numerically
The solution of a PDE involves the direct relationship between the depen-dent and independepen-dent variables and may be analytical or numerical. A PDE analytical solution is an actual mathematical expression describing the de-pendent variable as a function of indede-pendent variables [14]. However, such a solution is most easily obtained for linear equations and not for the (non-linear) system of equations described in this thesis. In a PDE numerical solution, the equations are simulated and thereafter the numerical values of the dependent variable are plotted as a function of space and time. For many realistic problems in science and engineering and including the one described in this thesis, only numerical solutions to (systems of) PDEs are available, due to the complexity of these PDEs, but they generally have good accuracy [14].
Several methods exist to numerically simulate a system of differential equa-tions and in this thesis we use method of lines (MOL). In MOL, all but one dimension are discretised, such that instead of solving a PDE over time
and space, many separate ODEs for separate points in space are solved
numerically over time only.
1.3 Thesis outline
other methods for solving PDEs [14].
2) The MOL procedure is highly flexible: the PDEs under study are re-placed with systems of approximating ODEs and numerous combinations of algebraic and differential equations are possible [14].
In MOL, to the purpose of approximating the original system of PDEs, lines are defined within the studied domain. The lines are distributed equally over space and a particular system of ODEs is then solved at each intersection of lines. In Figure 1.4 an image of how this looks for the 2D brain unit model is presented. While a higher amount of lines increases the accuracy of the model, it also increases the computational time. After weighing accuracy and computational cost, as well as taking into account that decreasing or increasing the amount of lines should not substantially affect the simulation results, an amount of 18 lines per dimension is chosen to proceed with in the simulations performed for this thesis. This means that both the single 3D brain unit (Chapter 4) and the 3D brain unit network are described by 18x18x18 lines. This corresponds to a simulation time in the order of hours.
Figure 1.4:Method of lines. The domain under study, in this case the 2D brain ECF (blue) surrounded by the blood of the brain capillaries (red), is discretized over space. This allows for the numerical simulation of multiple ODEs (one for each point in space) rather than one PDE coupled to two ODEs for the entire system.
1.3
Thesis outline
distri-drug distribution processes have been addressed in these models and which have not. We highlight the need for a more comprehensive and integrated model that fills the current gaps in predicting spatial drug distribution within the brain.
In Chapter 3, we set up a 2D basic model that describes drug distribution within the brain, covering the most essential aspects of drug distribution within the brain. We introduce the ‘2D brain unit’ and describe the change in concentration of unbound drug in this unit by the basic model given in section 1.2.1.1. We show how BBB permeability and drug binding kinetics affect the concentration-time profiles of free and bound drug. Moreover, we show that we are able to detect local differences in drug concentrations within the 2D brain unit.
In Chapter 4, we formulate a ‘3D brain unit’ that builds on the 2D model. In addition, we extend the model with a drug plasma domain (see 1.2.1.2) and descriptions of active BBB transport (see 1.2.1.3). The 3D brain unit model enables the study of many parameters, but we focus on the newly implemented parameters. We show that the brain capillary blood flow, pas-sive and active BBB transport affect subsequent drug distribution within the brain ECF in an interdependent manner.
In Chapter 5, we present a network of 3D brain units to understand how drug concentrations may differ locally and under different (normal and disease) conditions. We show that disease-induced changes in brain tis-sue characteristics may significantly affect drug concentrations within the brain ECF. The extent hereof depends on the drug properties.
